How to make group of vertices solid/frozen/rigid
Option 1:

Separate the objects. In "object mode" hit "shift+d" to duplicate the object. Delete the vertices in one object, and the inverse vertices in the other. Then, while editing one, you do not affect another.

Option 2:

Deform your mesh with an Armature. Use 2 bones. One bone affects the rigid piece, the other bone affects the other parts. Pose the bones into the shape you want, and apply the armature. Use "vertex groups" to influence the deforms. A bone will affect vertices in a vertex group that has the same name as it.

Hopefully this can help you learn some terminology and put you on the right track.